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Norm rigidity for arithmetic and profinite groups
Group Theory · Submitted 10 May 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact.
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The abstract, sentence by sentence
Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If $G$ is any group satisfying this dichotomy we say that $G$ has the \emph{dichotomy property}. We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. | background 0.99 | 0.06 | 0.02 | asserted 0.84 |
| s1 | We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. | result 0.98 | 0.95 | 0.89 | asserted 1.00 |
| s2 | If $G$ is any group satisfying this dichotomy we say that $G$ has the \emph{dichotomy property}. | background 0.39 | 0.60 | 0.05 | asserted 0.83 |
| s3 | We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. | contribution 0.57 | 0.77 | 0.06 | asserted 0.98 |
| s4 | As a consequence we derive constraints to the possible approximations of certain non residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. | result 0.63 | 0.94 | 0.09 | asserted 0.58 |
| s5 | In the last section we provide several open problems for further research. | future work 1.00 | 0.79 | 0.02 | asserted 0.88 |
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The paper
Original abstract
Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If $G$ is any group satisfying this dichotomy we say that $G$ has the \emph{dichotomy property}. We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research.
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How the literature uses it
1 arXiv paper cite this in 3 sentences
What it cites · 3 sentences without a confident label
Uses the method or tool of 1 paper
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Trost [ref] found a way around this issue, by adapting Lemma [ref].
The dichotomy property of ${\rm SL}_2(R)$-A short note · Questions · 0.98
Cites as background 1 paper
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Two especially challenging cases seem to stand out: The first is that when $$ is anisotropic over $K$, i.e. $Γ$ has no unipotent elements (and is a co-compact lattice in its “natural envelope”; see [ref] for more detalis, as well as a construction of such groups by means of maximal orders in division algebras).
Introduction to Arithmetic Groups · Questions · 1.00
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